Thank you, virtual6. I have decided what I'm going to do now with your help and the help of the others who have given me information in this thread.When townsbg said "USB drive", yes that could be an external hard drive. It could also refer to the very popular and convenient so-called flash drive (or key drive, thumb drive, etc.)...the modern replacement for the floppy disk.
You are correct about how to use the hard drive enclosure. But you need to know the specifics of your drive in order to select the correct enclosure. The one you linked may or may not work, it's for a 2.5 inch drive, but your tower may have a 3.5 inch drive. Since you're just going to keep the old computer around for a while, I wouldn't recommend to try to remove the old drive, buy an enclosure, etc.
Instead just get an external USB drive like mentioned above, easily available at local big box and office supply stores.
Plus you can use it for system backups, extra storage, etc. The flash drive capacity of course is quite a bit less than a hard drive, but might be enough for file transfers.